The origins of chess and related games are lost in time; yet, in spite of hundreds of years of analysis, they remain as interesting as ever, because of their fantastically large configuration space. The articles presented here are steps in the continuing endeavor to master these games, an endeavor in which the computer nowadays is often a valuable tool. In fact, the “simpler” board game Nine Men's Morris has succumbed to computer analysis, as reported by R. Gasser. Checkers may well be on its way: J. Schaeffer tells of the development of the program Chinook, and pays a tribute to the extraordinary (human!) player M. Tinsley. N. Elkies and L. Stiller write articles about chess, computerless in one case and computer-heavy in the other. Shogi, also called Japanese chess, is Y. Kawano's subject.

The last four articles of this section deal with Go, a game that has come under intense scrutiny recently. Although it is a territorial game and not, strictly speaking, a combinatorial game according to the definition on page 1, the board breaks up toward the end into a sum of smaller independent games, a situation that the theory of combinatorial games handles well. Other aspects of Go, such as ko, require extensions of the traditional theory, as explained in two of these articles.

1. Introduction

We continue our introduction to combinatorial games with a survey of impartial games. Most of this material can also be found in WW [Berlekamp et al. 1982], particularly pp. 81-116, and in ONAG [Conway 1976], particularly pp. 112-130. An elementary introduction is given in [Guy 1989]; see also [Fraenkel 1996a], pp. 13-42 in this volume.

This is the inductive step that proves the Sprague-Grundy theorem [Sprague 1935-36; Grundy 1939], which states that every position in an impartial game (or, which is the same, every impartial game) is equivalent to a nim-heap (see page 20ff. in Fraenkel's article in this volume).

2. Examples of Impartial Games

We all know that the game of Nim is played with several heaps of beans. A move is to select a heap, and to remove any positive number of beans from it, possibly the whole heap. Any position in Nim is therefore the sum of several one-heap Nim games. The value of a single heap of n beans is *n. It's easy to see how to win a game of Nim if there's only one (nonempty) heap: take the whole heap! But it's worth pausing for a moment to note exactly what your options are. They are to move to any smaller sized heap: they correspond exactly to the options in the definition of *n. It's also fairly easy to play well at two-heap Nim: if the heaps are unequal in size, remove enough beans from the larger to make the heaps equal.

1. Introduction

There are two game theories. The first is now sometimes referred to as matrix game theory and is the subject of the famous von Neumann and Morgenstern treatise [1944]. In matrix games, two players make simultaneous moves and a payment is made from one player to the other depending on the chosen moves. Optimal strategies often involve randomness and concealment of information. The other game theory is the combinatorial theory of Winning Ways [Berlekamp et al. 1982], with origins back in the work of Sprague [1936] and Grundy [1939] and largely expanded upon by Conway [1976]. In combinatorial games, two players move alternately. We may assume that each move consists of sliding a token from one vertex to another along an arc in a directed graph. A player who cannot move loses. There is no hidden information and there exist deterministic optimal strategies.

In the late 1980's, David Richman suggested a class of games that share some aspects of both sorts of game theory. Here is the set-up: The game is played by two players (Mr. Blue and Ms. Red), each of whom has some money. There is an underlying combinatorial game in which a token rests on a vertex of some finite directed graph.

1. Introduction

Connections between combinatorial games (simply games in the sequel) and linear error-correcting codes (codes in the sequel) have been established in [Conway and Sloane 1986; Conway 1990; Brualdi and Pless 1993], where lexicodes, and some of their connections to games, are explored. Our aim is to extend the connection between games and codes to a large class of games, and to formulate a potentially polynomial method for generating codes from games. We also establish the basic properties of lexicodes by a simple, transparent method.

Let Γ, any finite digraph, be the groundgraph on which we play the following general two-player game. Initially, distribute a positive finite number of tokens on the vertices of Γ. Multiple occupation is permitted. A move consists of selecting an occupied vertex and moving a single token from it to a neighboring vertex, occupied or not, along a directed edge.

Combinatorial Game Theory, as an academic discipline, is still in its infancy. Many analyses of individual games have appeared in print, starting in 1902 with C. L. Bouton's analysis of the game of Nim. (For exact references to the works mentioned here, please see A. Praenkel's bibliography on pages 493-537 of this volume.) It is was not until the 1930's that a consistent theory for impartial games was developed, independently, by R. Sprague and P. M. Grundy, later to be expanded and expounded upon by R. K. Guy and C. A. B. Smith. (Guy is still going strong, as evidenced by his energy at this Workshop.) J. H. Conway then developed the theory of partizan games, which represented a major advance. Based on this theory, D. Knuth wrote his modern moral tale, Surreal Numbers. The collaboration of E. R. Berlekamp, J. H. Conway and R. K. Guy gave us Winning Ways, which “set to music”, or at least to “rhyme”, the theory so far. In the process, many more games were discovered and analyzed: but more were discovered than solved!

This Workshop, held from 11 to 21 July 1994, gave evidence of the growing interest in combinatorial games on the part of experts from many fields: mathematicians, computer scientists, researchers in artificial intelligence, economists, and other social scientists. Players, some of whom make their living from games, also attended. Visitors such as D. Knuth and H. Wilf dropped in for a few hours or days and gave impromptu lectures. There was much cross-fertilization of ideas, as could be expected from a meeting of people from such varied backgrounds.

1. Introduction

The game called Kotzig's Nim in Winning Ways [Berlekamp et al. 1982] and Modular Nim in [Fraenkel et al. 1995] consists of a directed cycle of length n with the vertices labelled 0 through n — 1, a coin placed initially on vertex 0, and a set of integers called the move set. There are two players, who alternate moves; a move consists of moving the coin from the vertex i on which it currently resides to vertex i + m mod n, where m is a member of the move set. However, the coin can only land on a vertex once. Thus, the game is finite. The last player to move wins. Most of the known results concern themselves with move sets of small cardinality and consisting of small numbers (see p. 481 of Winning Ways, and [Fraenkel et al. 1995]).

Obviously, this game can be extended to more general directed graphs, the move set being indicated by directed edges, for clarity. This has become known as Geography [Fraenkel et al. 1993; Fraenkel and Simonson 1993]. The Grundy value of a game G, denoted g(G), is either P for a previous-player win or N for a next-player win. Throughout, we call the first player Algois and the second Berol.

This article skims the surface of the vast subject of combinatorial games. It often makes reference to the foundational books Winning Ways for Your Mathematical Plays, abbreviated WW [Berlekamp et al. 1982], and On Numbers and Games, abbreviated ONAG [Conway 1976]. Other references that should be consulted are [Praenkel 1980; Guy 1983; Guy 1991], and the other articles in this volume. See also Fraenkel's master bibliography on pages 493-537.

1. What We Mean by a Combinatorial Game

Our games are unlike those of “classical” game theory, that find application in economics, management, and military strategy. Our games usually, though perhaps not quite always, satisfy the following conditions:

• 1. There are just two players, often called Left and Right. There can be no question of coalitions.

• 2. There are several, usually finitely many, positions, and often a particular starting position.

• 3. There are clearly defined rules that specify the two sets of moves that Left and Right can make from a given position to its options.

• 4. Left and Right move alternately, in the game as a whole.

• 5. In the normal play convention a player unable to move loses.

• 6. The rules are such that play will always come to an end because some player will be unable to move. This is called the ending condition. There are no games that are drawn by repetition of moves.

• 7. Both players know what is going on; there is complete information. There is no occasion for bluffing.

• 8. There are no chance moves: no dealing of cards; no rolling of dice.

1. Introduction

In the two-player strategy game of Go, it can happen that an endgame position splits up into several non-interacting subpositions. Since each player must then move in just one of the subpositions on his turn, the whole position will then be the so-called disjunctive compound, or sum, of the subpositions. As it turns out, we can then apply to these Go endgames the theory of partizan combinatorial games with finite play under disjunctive composition, as found in Winning Ways [Berlekamp et al. 1982], Chapters 1-8, or On Numbers and Games [Conway 1976].

This paper assumes that the reader is already somewhat familiar with Go and with the application of this theory to Go, as given in [Wolfe 1991; Berlekamp and Wolfe 1994]. In Chapter 11 of Winning Ways there is a theory of partizan combinatorial games with possibly infinite play under disjunctive composition. These games are there called loopy, since what was a game tree in the finite play case is now a game graph, perhaps with cycles. We review this theory of loopy games and show how it can be applied to Go, which also has cycles.

1. What are Combinatorial Games? What are they Good For?

Roughly speaking, the family of combinatorial games consists of two-player games with perfect information (no hidden information as in some card games), no chance moves (no dice) and outcome restricted to (lose, win), (tie, tie) and (draw, draw) for the two players who move alternately. Tie is an end position such as in tic-tac-toe, where no player wins, whereas draw is a dynamic tie: any position from which a player has a nonlosing move, but cannot force a win. Both the easy game of Nim and the seemingly difficult chess are examples of combinatorial games. We use the shorter terms game and games below to designate combinatorial games.

Amusing oneself with games may sound like a frivolous occupation. But the fact is that the bulk of interesting and natural mathematical problems that are hardest in complexity classes beyond NP, such as Pspace, Exptime and Expspace, are two-player games; occasionally even one-player games (puzzles) or even zeroplayer games (Conway's “Life“). Two of the reasons for the high complexity of two-player games are outlined below. Before that we note that in addition to a natural appeal of the subject, there are applications or connections to various areas, including complexity, logic, graph and matroid theory, networks, errorcorrecting codes, surreal numbers, on-line algorithms and biology.

But when the chips are down, it is this “natural appeal” that compels both amateurs and professionals to become addicted to the subject. What is the essence of this appeal? Perhaps it is rooted in our primal beastly instincts; the desire to corner, torture, or at least dominate our peers.

Marion Tinsley died on April 3, 1995, at the age of 68. Why does the death of this checkers ( 8 x 8 draughts) player attract our attention? His record speaks for itself:

Since an accidental loss in the 1950 U.S. Championship, Tinsley finished in undivided first place in every tournament that he played in, except the last (in the 1994 U.S. Championship, he tied for first place with the computer program Chinook and Don Lafferty). He contested nine World Championship matches, winning each usually by an embarrassingly large margin. Over the last fortyfive years of his life, comprising thousands of tournament, World Championship, match, exhibition and casual games, Tinsley lost the unbelievable number of seven games. Seven games!? In forty-five years? This is as close to perfection as is humanly possible.

Tinsley once remarked that he had become bored playing humans; there wasn't any challenge left. When he was young, he began to acquire the reputation of being unbeatable. For forty-five years, most of his opponents would play for the draw; going for a win was unthinkable. Tinsley's enjoyment of checkers waned, and at one point he retired from the game for twelve years because of a lack of competition.

When the program Chinook came on the scene, Tinsley relished the opportunity to play it. Chinook had no respect for Tinsley's abilities, willingly taking risks: anything to increase the chances of winning. Tinsley said that playing Chinook made him feel like a young man again. In 1990, Chinook earned the right to play Tinsley for the (human) World Checkers Championship.

As in our earlier articles, WW stands for Winning Ways [Berlekamp et al. 1982]. We say that the nim-value of a position is n when its value is the nimber *n.

1. Subtraction games are known to be periodic. Investigate the relationship between the subtraction set and the length and structure of the period.

(For subtraction games, see WW, pp. 83-86, 487-498, and Section 4 of [Guy 1996] in this volume, especially the table on page 67. A move in the subtraction game S(s1,S2,S3,…) is to take a number of beans from a heap, provided that number is a member of the subtraction set ﹛ s1,S2,S3,…﹜. Analysis of such a game and of many other heap games is conveniently recorded by a nim-sequence, , meaning that the nim-value of a heap of h beans in this particular game is: in symbols, Arbitrarily difficult subtraction games can be devised by using infinite subtraction sets: the primes, for example.) The same question can be asked about partizan subtraction games, in which each player is assigned an individual subtraction set [Fraenkel and Kotzig 1987].

1. Introduction

Toads and Frogs is a two-player game, played on a one-dimensional board. Left has a number of toads, and Right has a number of frogs, each on its own square of the board. Each player has two types of legal moves: he may either push one of his pieces forward into an adjacent empty square, or he may jump one of his pieces over an adjacent opposing piece, into an empty square. Jumps are never forced, and jumped-over pieces are not affected in any way. Toads move to the right, frogs to the left. The first player without a legal move loses the game.

Throughout the paper, we represent toads by T, frogs by F, and empty squares by the symbol □.

Here is a typical Toads and Frogs game. Left moves first and wins.

Domineering is a game played on subsets of the square lattice by two players, who alternately remove connected two-square regions (dominoes) from play. Left may only place dominoes vertically; Right must play horizontally. The normal win-condition applies, so that the first player unable to move loses.

It is difficult to analyse general Domineering positions, even for quite small boards. One way to gain insight into the nature of the problem is to watch actual games between expert players. To determine who were the strongest players available, an open-registration tournament was held. To insure that the players gave proper consideration to their play, a prize of $500 was awarded the winner.

The finalists were Dan Calistrate of Calgary and David Wolfe of Berkeley. The format for the final was to play two games, each player taking the first turn once; if the series was split, two more games would be played, and so on until one player won both games of a round. As in chess, it would be expected that one set of pieces would provide an advantage, therefore winning with the favoured set would be like holding serve in tennis. In the event, the first round produced two first-player wins, and Calistrate won both second-round games.

The following article is based on unpublished work of the late mathematician David Ross Richman (1956-1991).

David was a problem solver by temperament, with strong interests in number theory, algebra, invariant theory, and combinatorics. He did his undergraduate work at Harvard and received his Ph.D. in mathematics from the University of California at Berkeley, under the supervision of Elwyn Berlekamp. I met him at one of the annual convocations of the West Coast Number Theory Conference held at the Asilomar Conference Center in Monterey, California. His quick mind and unassuming manner made him a pleasant person to discuss mathematics with, and he was one of the people I most looked forward to seeing at subsequent conferences.

In one of our conversations in the mid-1980's, he mentioned his idea of playing combinatorial games under a protocol in which players bid for the right to make the next move. Over the course of the next few years, I urged him to write up this work, but he was too busy with other mathematical projects.

By the beginning of 1991, he had received tenure at the University of South Carolina, and was commencing his first sabbatical. He planned to spend the first half of 1991 in Taiwan and the second half at MSRI. He died on February 1, 1991, in a widely-reported accident at Los Angeles Airport in which many other people were killed. He left behind a wife, the mathematician Shu-Mei Richman; a daughter, Miriam; and his parents, Alex and Shifra.

1. Introduction

Checkers is a popular game around the world, with over 150 documented variations. Only two versions, however, have a large international playing community. What is commonly known as checkers (or draughts) in North America is widely played in the United States and the British Commonwealth. It is played on an 8 x 8 board, with checkers moving one square forward and kings moving one square in any direction. Captures take place by jumping over an opposing piece, and a player is allowed to jump multiple men in one move. Checkers promote to kings when they advance to the last rank of the board. So-called international checkers is popular in the Netherlands and the former Soviet Union. This variant uses a 10 x 10 board, with checkers allowed to capture backwards, and kings allowed to move many squares in one direction, much like bishops in chess. This paper is restricted to the 8 x 8 variant, but many of the ideas presented here also apply to the 10 x 10 game.

1. The Simplest Form of the Game

Consider the following game, played by two players, Left and Right: Coins of various (positive numeric) monetary values, colored red or blue, are placed on a semi-infinite strip. We call the red coins Right's and the blue coins Left's. The playing field and possible moves are indicated in Figure 1. Each player can, on his turn, either slide one of his coins down one square, or remove one of his opponent's coins from the strip. Each player gets to keep all the money he moves off the bottom of the strip. The winner is the player who ends up with the most money, or, in the event of a tie, the player who moved last.

Since each player moves just one coin at each turn, the overall game is a disjunctive composition of subgames corresponding to the individual positions. In fact, we can assign a partizan combinatorial game, in the sense of Winning Ways [Berlekamp et al. 1982], Chapters 1-8, or On Numbers and Games [Conway 1976], to each coin on a square, and the overall game will then be the sum of these individual games, which we call terms. Figure 1 has names for these terms [Berkekamp and Wolfe 1994, §§4.1, 4.11; Wolfe 1994, §§3.1, 3.10]. We have also numbered our squares, with square 0 at the bottom.

1. Introduction

We will use the classical definitions and facts about two-person, perfect information combinatorial games with the normal winning convention, as developed in Winning Ways [Berlekamp et al. 1982] and On Numbers and Games [Conway 1976]. We recapitulate them briefly.

Formally, games are constructed recursively as ordered pairs ﹛ ΓL\ ΓR﹜, where ΓL and ΓR are sets of games, called, respectively, the set of Left options and the set of Right options from G. We will restrict ourselves to short games, that is, games where the sets of options ΓL and ΓR are required to be finite in this recursive definition. The basis for this recursion is the game ﹛∅ |∅ ﹜, which is called 0. We will often let GL and GR represent typical Left and Right options of a game G, and write G = ﹛GL | GR﹜.

Two games G and H are identical, or have the same form, if they have identical sets of left options and identical sets of right options. In this case we write G = H. Whenever the distinction between the value and the form of a game is essential, we will specify it; otherwise, by G we will mean the form of G.